Equivalence Relations

• Read carefully pages 196 through 198 (section 8.3)
• Some key questions to answer:
  1. When do we say that a relation R on set A is an equivalence relation?
  2. For any given set A there is a “smallest possible” equivalence relation. Think of what that relation would be. What are the ordered pairs we are forced to add to an equivalence relation, no matter what?
  3. For a given equivalence relation \(R\) on a set \(A\), and an element \(a\in A\), what is the equivalence class \([a]\)?
  4. What is the equivalence class for an integer \(n\) where the set \(A=\mathbb{Z}\) and the equivalence relation is integer equality?
  5. Suppose that \(A\) is a subset of the real numbers, such that the relation “less than or equal to” on \(A\) is an equivalence. Prove that \(A\) must have at most one element.
• Practice problems from section 8.3 (page 211): 8.25, 8.28, 8.29, 8.31, 8.34